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Definite integrals and area

  1. Feb 28, 2015 #1
    1. The problem statement, all variables and given/known data

    Find ∫ f(x) dx between [4,8]

    if,

    ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4

    2. Relevant Equations

    ∫ f(x) dx between [4,8] ,
    ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4
    pic.jpg

    3. The attempt at a solution

    We are given ∫ f(2x) dx between [1,4] = 3

    Let, 2x = u ⇒ dx = du/2

    So, the new intervals are u2 = 2(4) = 8 and u1 = 2(1) = 2

    This gives: 1/2 ∫ f(u) du between [2,8] = 3 ⇒ ∫ f(u) du between [2,8] = 6

    And so to find t ∫ f(x) dx between [4,8]

    I subtract ∫ f(x) dx between [2,4] from ∫ f(2x) dx between [1,4]

    which is the same as writing: ∫ f(u) du between [2,8] - ∫ f(x) dx between [2,4] = 6- 4 = 2

    Is this the correct answer? I'm not sure if ∫ f(u) du between [2,8] = 6 is the same as ∫ f(x) dx between [2,8] = 6
    I read something about a dummy variable and this seems like a reasonable answer. Please let me know.
     
  2. jcsd
  3. Feb 28, 2015 #2

    wabbit

    User Avatar
    Gold Member

    Yes your reasoning and result are correct. And indeed x or u are just placeholders in the integrals, you can replace them with any symbol you like.
     
  4. Feb 28, 2015 #3
    thank you
     
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